Stanford CS234 Reinforcement Learning I Policy Evaluation I 2024 I Lecture 3

01 Nov 2024 (2 months ago)

A poll is conducted to refresh understanding of Markov Decision Processes (MDPs), focusing on their properties and guarantees.

A poll is conducted to refresh understanding of Markov Decision Processes (MDPs), focusing on their properties and guarantees. (5s)

Tabular MDPs are discussed, where the value of each state can be represented in a table, contrasting with neural networks that do not have one parameter per state.

Tabular MDPs are discussed, where the value of each state can be represented in a table, contrasting with neural networks that do not have one parameter per state. (51s)

The concept of (a^s) is explained as the number of possible deterministic policies, where (a) is the number of actions and (s) is the number of states.

The concept of (a^s) is explained as the number of possible deterministic policies, where (a) is the number of actions and (s) is the number of states. (2m24s)

It is confirmed that value iteration and policy iteration both asymptotically converge and compute the correct value function in tabular discrete MDPs.

It is confirmed that value iteration and policy iteration both asymptotically converge and compute the correct value function in tabular discrete MDPs. (3m0s)

A discussion is held on the convergence of value iteration, which is guaranteed if the discount factor (\gamma) is less than one, although it may require more iterations compared to policy iteration.

A discussion is held on the convergence of value iteration, which is guaranteed if the discount factor (\gamma) is less than one, although it may require more iterations compared to policy iteration. (5m16s)

An example is given to illustrate the difference between policy iteration and value iteration, using a simple MDP with one state and one action, where policy iteration takes one round, but value iteration continues until the value function stabilizes.

An example is given to illustrate the difference between policy iteration and value iteration, using a simple MDP with one state and one action, where policy iteration takes one round, but value iteration continues until the value function stabilizes. (6m25s)

The discussion involves using a geometric series to evaluate the value function in reinforcement learning, where the value of a state is calculated as (1/(1-\gamma)), with (\gamma) being the discount factor. This reflects the idea of staying in the same state and taking the same action repeatedly.

The discussion involves using a geometric series to evaluate the value function in reinforcement learning, where the value of a state is calculated as (1/(1-\gamma)), with (\gamma) being the discount factor. This reflects the idea of staying in the same state and taking the same action repeatedly. (7m5s)

In value iteration, the initial value (V1(s)) is 1, which is not close to the eventual value of 10, indicating that multiple iterations are needed for convergence. In contrast, policy iteration would stop after evaluating the value of the single policy, highlighting different short-term behaviors of these algorithms despite both eventually converging.

In value iteration, the initial value (V1(s)) is 1, which is not close to the eventual value of 10, indicating that multiple iterations are needed for convergence. In contrast, policy iteration would stop after evaluating the value of the single policy, highlighting different short-term behaviors of these algorithms despite both eventually converging. (7m32s)

The focus is on model-free policy evaluation in a simple setting without function approximation, where the agent learns by trying actions in the world without a given dynamics or reward model. This is done in a tabular setting where values for each state can be written separately.

The focus is on model-free policy evaluation in a simple setting without function approximation, where the agent learns by trying actions in the world without a given dynamics or reward model. This is done in a tabular setting where values for each state can be written separately. (8m34s)

Logistics for the course include office hours listed on a Google Calendar, with specific times for project and conceptual questions. Students are encouraged to discuss projects, which can involve new ideas, applications, or replicating existing research in reinforcement learning.

Logistics for the course include office hours listed on a Google Calendar, with specific times for project and conceptual questions. Students are encouraged to discuss projects, which can involve new ideas, applications, or replicating existing research in reinforcement learning. (9m3s)

The session aims to explore how to learn through direct interaction with the environment, continuing the theme of policy evaluation.

The session aims to explore how to learn through direct interaction with the environment, continuing the theme of policy evaluation. (10m21s)

The discussion focuses on evaluating the effectiveness of decisions under a fixed policy, using data from the environment obtained by executing that policy. This approach is crucial for making decisions and improving policies in complex algorithms like deep Q-learning and policy gradient methods.

The discussion focuses on evaluating the effectiveness of decisions under a fixed policy, using data from the environment obtained by executing that policy. This approach is crucial for making decisions and improving policies in complex algorithms like deep Q-learning and policy gradient methods. (10m31s)

The session aims to cover topics such as multicolor policy evaluation, temporal difference learning, certainty equivalence, and batch policy evaluation. Temporal difference learning and Q-learning are highlighted, with Q-learning being the control version of temporal difference learning.

The session aims to cover topics such as multicolor policy evaluation, temporal difference learning, certainty equivalence, and batch policy evaluation. Temporal difference learning and Q-learning are highlighted, with Q-learning being the control version of temporal difference learning. (11m43s)

Definitions are provided for key concepts: the return (G) as the discounted sum of rewards from a state, the state value function as the average reward, and the state-action value as the expected discounted sum of rewards when starting from a state and taking a specific action.

Definitions are provided for key concepts: the return (G) as the discounted sum of rewards from a state, the state value function as the average reward, and the state-action value as the expected discounted sum of rewards when starting from a state and taking a specific action. (12m8s)

Dynamic programming for policy evaluation is discussed, particularly when models are available, using Bellman-like backups for a specific policy. This differs from the Bellman equation as it does not involve maximizing over actions but follows the policy's specified actions.

Dynamic programming for policy evaluation is discussed, particularly when models are available, using Bellman-like backups for a specific policy. This differs from the Bellman equation as it does not involve maximizing over actions but follows the policy's specified actions. (12m35s)

The concept of bootstrapping is explained, where an estimate of the value function is used to calculate the expected discounted sum of future rewards. This involves using an immediate reward plus the discounted sum of future rewards as an estimate for further calculations.

The concept of bootstrapping is explained, where an estimate of the value function is used to calculate the expected discounted sum of future rewards. This involves using an immediate reward plus the discounted sum of future rewards as an estimate for further calculations. (14m8s)

Monte Carlo policy evaluation is a straightforward and widely used method in reinforcement learning, where the idea is to simulate or act in the real world using a given policy to determine the value function by averaging the returns from multiple episodes.

Monte Carlo policy evaluation is a straightforward and widely used method in reinforcement learning, where the idea is to simulate or act in the real world using a given policy to determine the value function by averaging the returns from multiple episodes. (14m19s)

The focus is primarily on deterministic policies, which simplifies the process by avoiding the need to calculate expectations over actions. The value function is essentially the mean of the returns obtained by following the policy.

The focus is primarily on deterministic policies, which simplifies the process by avoiding the need to calculate expectations over actions. The value function is essentially the mean of the returns obtained by following the policy. (14m39s)

An example of Monte Carlo policy evaluation is assessing the average effectiveness of a medical treatment by applying it to multiple patients and averaging their outcomes. This method involves executing the policy across various episodes and averaging the results.

An example of Monte Carlo policy evaluation is assessing the average effectiveness of a medical treatment by applying it to multiple patients and averaging their outcomes. This method involves executing the policy across various episodes and averaging the results. (15m33s)

The method can handle cases where trajectories have different lengths, such as when patients drop out of a trial or complete their treatment at different times. The value function is derived by averaging over all trajectories, regardless of their length.

The method can handle cases where trajectories have different lengths, such as when patients drop out of a trial or complete their treatment at different times. The value function is derived by averaging over all trajectories, regardless of their length. (16m5s)

Monte Carlo policy evaluation does not assume that the system is a Markov decision process (MDP), meaning it can be applied even if the system's state is not fully described by the available features. It simply involves averaging the outcomes of multiple policy rollouts.

Monte Carlo policy evaluation does not assume that the system is a Markov decision process (MDP), meaning it can be applied even if the system's state is not fully described by the available features. It simply involves averaging the outcomes of multiple policy rollouts. (16m42s)

This approach is applicable only to episodic MDPs, where episodes have a defined end, allowing for the calculation of total returns. It is not suitable for continuous, never-ending processes without additional approximations.

This approach is applicable only to episodic MDPs, where episodes have a defined end, allowing for the calculation of total returns. It is not suitable for continuous, never-ending processes without additional approximations. (17m33s)

The discussion addresses whether a treatment can be considered episodic if the episodes have different lengths, concluding that it can be episodic as long as the episode is guaranteed to end, allowing for averaging over outcomes.

The discussion addresses whether a treatment can be considered episodic if the episodes have different lengths, concluding that it can be episodic as long as the episode is guaranteed to end, allowing for averaging over outcomes. (18m26s)

To evaluate the value function for all states, it is necessary to observe all states within the trajectories, and methods for estimating these values are discussed.

To evaluate the value function for all states, it is necessary to observe all states within the trajectories, and methods for estimating these values are discussed. (19m4s)

The process involves initializing variables to count the number of times a state has been updated and to track the returns from each state, using the every-visit Monte Carlo method to sample episodes and compute the discounted sum of rewards.

The process involves initializing variables to count the number of times a state has been updated and to track the returns from each state, using the every-visit Monte Carlo method to sample episodes and compute the discounted sum of rewards. (19m26s)

The first-visit Monte Carlo evaluation method updates a state at most once per episode, even if the state is visited multiple times, which can be problematic if a state is rarely visited, leading to undefined values for such states.

The first-visit Monte Carlo evaluation method updates a state at most once per episode, even if the state is visited multiple times, which can be problematic if a state is rarely visited, leading to undefined values for such states. (21m0s)

Estimating rare side effects of treatment plans, such as those from the COVID vaccine, requires a large number of trajectories because some side effects only appear after many observations. This highlights the challenge of needing numerous episodes to observe infrequent states.

Estimating rare side effects of treatment plans, such as those from the COVID vaccine, requires a large number of trajectories because some side effects only appear after many observations. This highlights the challenge of needing numerous episodes to observe infrequent states. (22m1s)

In reinforcement learning, the "every visit" method updates the value estimate every time a state is encountered in a trajectory, which can be beneficial for long trajectories where a state is visited multiple times.

In reinforcement learning, the "every visit" method updates the value estimate every time a state is encountered in a trajectory, which can be beneficial for long trajectories where a state is visited multiple times. (22m54s)

The "incremental Monte Carlo" method maintains a running estimate of the value under a policy for a particular state, updating it smoothly as more data is collected. This involves weighing the old estimate by the number of times the state has been visited and incorporating the new return observed.

The "incremental Monte Carlo" method maintains a running estimate of the value under a policy for a particular state, updating it smoothly as more data is collected. This involves weighing the old estimate by the number of times the state has been visited and incorporating the new return observed. (23m35s)

The incremental update process is similar to a learning rate in machine learning, where the updated value is a combination of the old value and the new data. This method allows for a flexible learning rate, denoted as Alpha, to update the value function for a state.

The incremental update process is similar to a learning rate in machine learning, where the updated value is a combination of the old value and the new data. This method allows for a flexible learning rate, denoted as Alpha, to update the value function for a state. (24m21s)

The estimates used in these methods, often referred to as targets, involve calculating the return starting from a specific state until the end of the episode. This concept is visually similar to structures like Mini Max trees or Expecting Max trees, which are used in algorithms such as AlphaGo.

The estimates used in these methods, often referred to as targets, involve calculating the return starting from a specific state until the end of the episode. This concept is visually similar to structures like Mini Max trees or Expecting Max trees, which are used in algorithms such as AlphaGo. (25m0s)

The discussion focuses on policy evaluation in reinforcement learning, where the goal is to determine the value of starting in a certain state and taking a specific action as prescribed by a fixed policy. This involves considering the stochastic nature of the environment and the probabilities of reaching various next states.

The discussion focuses on policy evaluation in reinforcement learning, where the goal is to determine the value of starting in a certain state and taking a specific action as prescribed by a fixed policy. This involves considering the stochastic nature of the environment and the probabilities of reaching various next states. (25m39s)

Policy evaluation is conceptualized as a tree structure, where each state leads to a fixed action, and the branching factor represents all possible next states. The process involves calculating the expected value over all potential future states and propagating these values back to the root.

Policy evaluation is conceptualized as a tree structure, where each state leads to a fixed action, and the branching factor represents all possible next states. The process involves calculating the expected value over all potential future states and propagating these values back to the root. (26m52s)

The aim is to compute the value function in a computationally and sample-efficient manner. Monte Carlo policy evaluation is introduced as a method to approximate full expectations by using samples of returns, which are averaged over many iterations to estimate the value function.

The aim is to compute the value function in a computationally and sample-efficient manner. Monte Carlo policy evaluation is introduced as a method to approximate full expectations by using samples of returns, which are averaged over many iterations to estimate the value function. (27m55s)

Monte Carlo methods are highlighted for their ability to approximate the branching tree of possibilities by sampling, with more samples leading to better approximations. This approach has been foundational in the development of algorithms like Monte Carlo tree search, which was instrumental in advancements such as AlphaGo.

Monte Carlo methods are highlighted for their ability to approximate the branching tree of possibilities by sampling, with more samples leading to better approximations. This approach has been foundational in the development of algorithms like Monte Carlo tree search, which was instrumental in advancements such as AlphaGo. (28m41s)

Monte Carlo policy evaluation involves rolling out samples to estimate the value of a policy, and a key question is how good this estimate is. Various algorithms exist for policy evaluation, and important properties to consider include consistency, computational efficiency, and statistical efficiency. Consistency ensures that as more data is gathered, the estimate converges to the true value of the policy for all states.

Monte Carlo policy evaluation involves rolling out samples to estimate the value of a policy, and a key question is how good this estimate is. Various algorithms exist for policy evaluation, and important properties to consider include consistency, computational efficiency, and statistical efficiency. Consistency ensures that as more data is gathered, the estimate converges to the true value of the policy for all states. (29m14s)

Computational efficiency refers to the cost of computation and memory usage, while statistical efficiency relates to how the accuracy of the estimate changes with the amount of data. Empirical accuracy, such as mean squared error, is also important in evaluating the quality of an estimator.

Computational efficiency refers to the cost of computation and memory usage, while statistical efficiency relates to how the accuracy of the estimate changes with the amount of data. Empirical accuracy, such as mean squared error, is also important in evaluating the quality of an estimator. (30m26s)

The bias of an estimator is the difference between the average estimate and the true value, while variance is the difference between the estimate and its expectation squared. The mean squared error combines variance and bias squared. A desirable estimator has low mean squared error, low bias, and low variance.

The bias of an estimator is the difference between the average estimate and the true value, while variance is the difference between the estimate and its expectation squared. The mean squared error combines variance and bias squared. A desirable estimator has low mean squared error, low bias, and low variance. (31m2s)

An unbiased estimator is not necessarily consistent. Consistency means that as the amount of data increases, the probability that the estimate differs from the true value by more than a small amount approaches zero. Monte Carlo methods are generally unbiased and consistent, as they converge to the true value with an infinite amount of data.

An unbiased estimator is not necessarily consistent. Consistency means that as the amount of data increases, the probability that the estimate differs from the true value by more than a small amount approaches zero. Monte Carlo methods are generally unbiased and consistent, as they converge to the true value with an infinite amount of data. (31m55s)

First-visit Monte Carlo is unbiased and consistent, as it treats all data as independent and identically distributed (IID). In contrast, every-visit Monte Carlo is biased, as it does not treat data as IID.

First-visit Monte Carlo is unbiased and consistent, as it treats all data as independent and identically distributed (IID). In contrast, every-visit Monte Carlo is biased, as it does not treat data as IID. (32m21s)

Returns from visiting the same state multiple times in a trajectory are correlated, which means they are not independent and identically distributed (IID). This correlation can introduce bias, but it is consistent and often results in a lower mean squared error because more data is used within a single trajectory for updates.

Returns from visiting the same state multiple times in a trajectory are correlated, which means they are not independent and identically distributed (IID). This correlation can introduce bias, but it is consistent and often results in a lower mean squared error because more data is used within a single trajectory for updates. (33m5s)

Incremental Monte Carlo methods rely on the learning rate, denoted as Alpha, which balances new and old estimates. The learning rate can change over time, and if the sum of its values for a state approaches infinity while the sum of its squares remains finite, the method will converge to the true value. These conditions are common but not always necessary, and their applicability can depend on the problem domain.

Incremental Monte Carlo methods rely on the learning rate, denoted as Alpha, which balances new and old estimates. The learning rate can change over time, and if the sum of its values for a state approaches infinity while the sum of its squares remains finite, the method will converge to the true value. These conditions are common but not always necessary, and their applicability can depend on the problem domain. (33m36s)

The method is a high variance estimator, particularly with first-visit Monte Carlo, where a state is updated at most once per episode. This can lead to slow convergence, especially if outcomes from the same starting state vary significantly. The method is unbiased and consistent but requires episodic settings, meaning updates occur only at the end of an episode. This can be limiting in control and decision-making scenarios where immediate data use is beneficial.

The method is a high variance estimator, particularly with first-visit Monte Carlo, where a state is updated at most once per episode. This can lead to slow convergence, especially if outcomes from the same starting state vary significantly. The method is unbiased and consistent but requires episodic settings, meaning updates occur only at the end of an episode. This can be limiting in control and decision-making scenarios where immediate data use is beneficial. (34m59s)

The approach does not use the Markov process but updates the value function estimate using a sample of the return to approximate the expectation. Under mild conditions, it converges to the true value of the state. Even with knowledge of the true dynamics model and reward, this method can still be advantageous.

The approach does not use the Markov process but updates the value function estimate using a sample of the return to approximate the expectation. Under mild conditions, it converges to the true value of the state. Even with knowledge of the true dynamics model and reward, this method can still be advantageous. (36m15s)

Temporal difference learning is introduced as a key concept in reinforcement learning, related to Q-learning, which will be covered in a subsequent lecture. It is highlighted as central and novel to reinforcement learning, according to the textbook by Sutton and Barto.

Temporal difference learning is introduced as a key concept in reinforcement learning, related to Q-learning, which will be covered in a subsequent lecture. It is highlighted as central and novel to reinforcement learning, according to the textbook by Sutton and Barto. (36m55s)

Temporal difference learning is model-free, meaning it does not require a parametric representation of the reward function or dynamics model. It can be used in both episodic settings and infinite discounted horizon settings.

Temporal difference learning is model-free, meaning it does not require a parametric representation of the reward function or dynamics model. It can be used in both episodic settings and infinite discounted horizon settings. (38m42s)

The method involves updating estimates of the value of a state immediately after each state-action-reward-next state tuple, aiming to compute the expected discounted sum of rewards for a particular policy.

The method involves updating estimates of the value of a state immediately after each state-action-reward-next state tuple, aiming to compute the expected discounted sum of rewards for a particular policy. (39m1s)

The approach combines sampling to approximate expectations and bootstrapping to approximate future returns. Instead of using the full return, it uses the immediate reward plus the discounted sum of future rewards, leveraging the current estimate of the value function for the next state.

The approach combines sampling to approximate expectations and bootstrapping to approximate future returns. Instead of using the full return, it uses the immediate reward plus the discounted sum of future rewards, leveraging the current estimate of the value function for the next state. (38m37s)

In reinforcement learning, the value of the current state can be updated immediately upon reaching the next state, S Prime, without waiting until the end of the episode. This approach is applicable to infinite horizon problems.

In reinforcement learning, the value of the current state can be updated immediately upon reaching the next state, S Prime, without waiting until the end of the episode. This approach is applicable to infinite horizon problems. (40m35s)

The concept of the TD (Temporal Difference) Target is introduced, which involves adjusting the old estimate of a state's value by a learning rate towards a target. This target is the sum of the immediate reward and the discounted estimate of future rewards. The difference between the current estimate and the target is referred to as the TD0 error.

The concept of the TD (Temporal Difference) Target is introduced, which involves adjusting the old estimate of a state's value by a learning rate towards a target. This target is the sum of the immediate reward and the discounted estimate of future rewards. The difference between the current estimate and the target is referred to as the TD0 error. (41m0s)

The TD0 learning algorithm involves sampling a tuple of state, action, reward, and next state, updating the value for the starting state, and repeating this process iteratively. This is similar to Q-learning but without the use of a maximum function.

The TD0 learning algorithm involves sampling a tuple of state, action, reward, and next state, updating the value for the starting state, and repeating this process iteratively. This is similar to Q-learning but without the use of a maximum function. (41m54s)

An example is provided where a policy always takes action A1, with a discount factor of one. In this scenario, any action from state S1 or S7 terminates the episode. The trajectory starts in state S3, taking action A1, receiving a reward of zero, and transitioning to state S2.

An example is provided where a policy always takes action A1, with a discount factor of one. In this scenario, any action from state S1 or S7 terminates the episode. The trajectory starts in state S3, taking action A1, receiving a reward of zero, and transitioning to state S2. (42m33s)

The update process involves adjusting the value of state S3 using the formula: old estimate of V(S3) * (1 - Alpha) + Alpha * (immediate reward + gamma * V(S2)). Initially, all state values are set to zero, and a state's value will only change if it receives a non-zero immediate reward or transitions to a state with a non-zero value.

The update process involves adjusting the value of state S3 using the formula: old estimate of V(S3) * (1 - Alpha) + Alpha * (immediate reward + gamma * V(S2)). Initially, all state values are set to zero, and a state's value will only change if it receives a non-zero immediate reward or transitions to a state with a non-zero value. (43m13s)

In the given example, the only state that would be updated to a non-zero value is state S1, as it provides a reward of one. This update occurs when transitioning to state S1, highlighting the importance of non-zero rewards or transitions to states with non-zero values for updates.

In the given example, the only state that would be updated to a non-zero value is state S1, as it provides a reward of one. This update occurs when transitioning to state S1, highlighting the importance of non-zero rewards or transitions to states with non-zero values for updates. (45m4s)

The value function is updated using Temporal Difference (TD) learning, which updates after every state-action-reward-next state tuple, resulting in a different value function estimate compared to Monte Carlo methods.

The value function is updated using Temporal Difference (TD) learning, which updates after every state-action-reward-next state tuple, resulting in a different value function estimate compared to Monte Carlo methods. (45m25s)

In TD learning, the value of a terminal state is always zero, and the update involves a combination of the current estimate and the observed reward, weighted by a factor alpha.

In TD learning, the value of a terminal state is always zero, and the update involves a combination of the current estimate and the observed reward, weighted by a factor alpha. (45m39s)

Monte Carlo methods wait until the end of an episode to update the value function using the returns from all visited states, which can lead to different estimates compared to TD learning.

Monte Carlo methods wait until the end of an episode to update the value function using the returns from all visited states, which can lead to different estimates compared to TD learning. (46m47s)

The choice between TD and Monte Carlo methods can affect data efficiency, especially when data is limited at the beginning of learning. Sample efficiency is an important consideration in reinforcement learning.

The choice between TD and Monte Carlo methods can affect data efficiency, especially when data is limited at the beginning of learning. Sample efficiency is an important consideration in reinforcement learning. (47m14s)

TD learning approximates the expected value by sampling the next state and bootstraps using the current value estimate, unlike Monte Carlo methods that sample the entire value function.

TD learning approximates the expected value by sampling the next state and bootstraps using the current value estimate, unlike Monte Carlo methods that sample the entire value function. (47m40s)

The learning rate in TD learning affects how much the TD target is weighted compared to the past value estimate, and this has implications for convergence, especially in stochastic state spaces.

The learning rate in TD learning affects how much the TD target is weighted compared to the past value estimate, and this has implications for convergence, especially in stochastic state spaces. (48m44s)

A deterministic Markov Decision Process (MDP) is described, where the probability of transitioning to a specific next state given a state and action is 1, meaning there is no stochasticity in the transitions.

A deterministic Markov Decision Process (MDP) is described, where the probability of transitioning to a specific next state given a state and action is 1, meaning there is no stochasticity in the transitions. (49m25s)

Participants are encouraged to think about which statements are true regarding the deterministic MDP and to discuss their thoughts with others.

Participants are encouraged to think about which statements are true regarding the deterministic MDP and to discuss their thoughts with others. (49m48s)

A discussion takes place about the implications of different values of Alpha in the context of Temporal Difference (TD) learning. If Alpha equals zero, the TD target is ignored, and no updates occur. If Alpha equals one, the estimate is completely updated with each observation.

A discussion takes place about the implications of different values of Alpha in the context of Temporal Difference (TD) learning. If Alpha equals zero, the TD target is ignored, and no updates occur. If Alpha equals one, the estimate is completely updated with each observation. (55m59s)

An example is given to illustrate stochastic systems, where two states may transition to each other with certain probabilities, such as a 50% chance of receiving a reward of +1 or -1, highlighting the challenges of using a single sample to approximate expectations in stochastic environments.

An example is given to illustrate stochastic systems, where two states may transition to each other with certain probabilities, such as a 50% chance of receiving a reward of +1 or -1, highlighting the challenges of using a single sample to approximate expectations in stochastic environments. (56m41s)

It is noted that deterministic systems can still converge even if Alpha equals one, especially in cases with terminal states where no further transitions occur, such as reaching a terminal state with a fixed reward.

It is noted that deterministic systems can still converge even if Alpha equals one, especially in cases with terminal states where no further transitions occur, such as reaching a terminal state with a fixed reward. (58m4s)

In cases where there is no stochasticity towards the end of an episode, convergence can occur.

In cases where there is no stochasticity towards the end of an episode, convergence can occur. (58m33s)

Temporal Difference (TD) learning involves bootstrapping and sampling to approximate expectations, which can be used in both episodic and infinite horizon settings. It generally has lower variance and is a consistent estimator if the learning rate conditions are met.

Temporal Difference (TD) learning involves bootstrapping and sampling to approximate expectations, which can be used in both episodic and infinite horizon settings. It generally has lower variance and is a consistent estimator if the learning rate conditions are met. (58m57s)

TD(0) is introduced, which involves taking the immediate reward and bootstrapping by plugging in the value of the next state, as opposed to summing all discounted rewards for the entire episode.

TD(0) is introduced, which involves taking the immediate reward and bootstrapping by plugging in the value of the next state, as opposed to summing all discounted rewards for the entire episode. (59m28s)

There are various TD methods that interpolate between taking one step and plugging in the value versus not using any value function approximation.

There are various TD methods that interpolate between taking one step and plugging in the value versus not using any value function approximation. (1h0m19s)

The choice between computational complexity and performance often depends on the domain, and there is flexibility in using methods that balance these factors.

The choice between computational complexity and performance often depends on the domain, and there is flexibility in using methods that balance these factors. (1h0m58s)

Some methods do not require the Markov assumption, allowing for flexibility in systems where this assumption may not hold, often resulting in lower bias.

Some methods do not require the Markov assumption, allowing for flexibility in systems where this assumption may not hold, often resulting in lower bias. (1h1m6s)

The discussion also relates these ideas to dynamic programming, which can be used for policy evaluation when models are given.

The discussion also relates these ideas to dynamic programming, which can be used for policy evaluation when models are given. (1h1m43s)

Certainty equivalence approaches involve using data obtained from executing a policy to estimate a reward or dynamics model, such as a maximum likelihood Markov Decision Process (MDP) model, especially in a tabular setting.

Certainty equivalence approaches involve using data obtained from executing a policy to estimate a reward or dynamics model, such as a maximum likelihood Markov Decision Process (MDP) model, especially in a tabular setting. (1h2m10s)

A maximum likelihood estimate of the dynamics model can be obtained by counting the number of times a specific state-action pair transitions to a next state, divided by the number of times the state-action pair occurs. This approach can also be applied to the reward model and can be extended to use complex function approximators like deep neural networks. This model is referred to as a certainty equivalence model, which ignores errors in the models due to finite data.

A maximum likelihood estimate of the dynamics model can be obtained by counting the number of times a specific state-action pair transitions to a next state, divided by the number of times the state-action pair occurs. This approach can also be applied to the reward model and can be extended to use complex function approximators like deep neural networks. This model is referred to as a certainty equivalence model, which ignores errors in the models due to finite data. (1h2m30s)

Once the maximum likelihood Markov Decision Process (MDP) model is established, the value of a policy can be computed using existing methods, as both the dynamics and reward models are available. This approach is data-efficient because it updates the dynamics and reward models for all states and actions, allowing for the propagation of rewards across states. However, it is computationally expensive due to the need for full model policy evaluation.

Once the maximum likelihood Markov Decision Process (MDP) model is established, the value of a policy can be computed using existing methods, as both the dynamics and reward models are available. This approach is data-efficient because it updates the dynamics and reward models for all states and actions, allowing for the propagation of rewards across states. However, it is computationally expensive due to the need for full model policy evaluation. (1h3m11s)

The method is consistent and will converge to the correct solution for Markov models. It can be used for all policy evaluations, although it is computationally intensive. The term "NSA" refers to the number of times a state-action pair has been encountered.

The method is consistent and will converge to the correct solution for Markov models. It can be used for all policy evaluations, although it is computationally intensive. The term "NSA" refers to the number of times a state-action pair has been encountered. (1h4m26s)

The approach is different from Monte Carlo methods, as it involves generating models and propagating information rather than directly calculating returns. This distinction will lead to different decision-making processes.

The approach is different from Monte Carlo methods, as it involves generating models and propagating information rather than directly calculating returns. This distinction will lead to different decision-making processes. (1h5m15s)

Batch policy evaluation involves using a set of episodes to extract as much information as possible from the available data. This is particularly relevant in fields where data collection is expensive or potentially harmful, such as patient, student, or legal data.

Batch policy evaluation involves using a set of episodes to extract as much information as possible from the available data. This is particularly relevant in fields where data collection is expensive or potentially harmful, such as patient, student, or legal data. (1h5m33s)

The process of policy evaluation using a finite amount of data involves repeatedly sampling episodes and applying Monte Carlo or TD(0) methods to evaluate the policy. This process is iterated multiple times to understand the convergence of these methods.

The process of policy evaluation using a finite amount of data involves repeatedly sampling episodes and applying Monte Carlo or TD(0) methods to evaluate the policy. This process is iterated multiple times to understand the convergence of these methods. (1h6m13s)

An example is provided with a small domain consisting of two states, where the discount factor gamma is set to one, indicating no discounting. There are eight episodes of experience, with different trajectories and rewards observed.

An example is provided with a small domain consisting of two states, where the discount factor gamma is set to one, indicating no discounting. There are eight episodes of experience, with different trajectories and rewards observed. (1h6m44s)

In one scenario, starting in state A leads to a reward of zero, transitioning to state B with another reward of zero. In another scenario, starting in state B results in an immediate reward of one, observed six times, and once with a reward of zero.

In one scenario, starting in state A leads to a reward of zero, transitioning to state B with another reward of zero. In another scenario, starting in state B results in an immediate reward of one, observed six times, and once with a reward of zero. (1h6m58s)

When applying TD learning, the updates involve a combination of the old estimate and the immediate reward, with no future discounted rewards since the process terminates after state B. Iterating over this data leads to an estimated value of 0.75 for state B.

When applying TD learning, the updates involve a combination of the old estimate and the immediate reward, with no future discounted rewards since the process terminates after state B. Iterating over this data leads to an estimated value of 0.75 for state B. (1h7m33s)

For Monte Carlo evaluation, the process involves averaging all returns starting from state B. The question is whether the Monte Carlo method will converge to the same value as TD learning. The Monte Carlo estimate also results in a value of 0.75 for state B, as it averages the observed returns.

For Monte Carlo evaluation, the process involves averaging all returns starting from state B. The question is whether the Monte Carlo method will converge to the same value as TD learning. The Monte Carlo estimate also results in a value of 0.75 for state B, as it averages the observed returns. (1h9m21s)

The discussion involves evaluating the convergence of value functions ( V(a) ) using two different methods: Monte Carlo updates and Temporal Difference (TD) learning, in scenarios with finite data.

The discussion involves evaluating the convergence of value functions ( V(a) ) using two different methods: Monte Carlo updates and Temporal Difference (TD) learning, in scenarios with finite data. (1h11m4s)

The question posed is whether these methods compute the same value when data is limited, as opposed to having infinite data where both methods can be consistent.

The question posed is whether these methods compute the same value when data is limited, as opposed to having infinite data where both methods can be consistent. (1h11m40s)

In the example discussed, TD learning converges to a value of 75 for ( V(a) ), based on the immediate reward being zero and the discounted reward from the next state, which was previously calculated as 75.

In the example discussed, TD learning converges to a value of 75 for ( V(a) ), based on the immediate reward being zero and the discounted reward from the next state, which was previously calculated as 75. (1h15m12s)

Monte Carlo, on the other hand, does not converge to 75; instead, it converges to zero because it averages over all observed returns, and in this case, only one trajectory was observed where ( a ) appeared, resulting in a return of zero.

Monte Carlo, on the other hand, does not converge to 75; instead, it converges to zero because it averages over all observed returns, and in this case, only one trajectory was observed where ( a ) appeared, resulting in a return of zero. (1h16m45s)

It is highlighted that while both methods converge to the correct value asymptotically under certain assumptions, with finite data, they may converge to different values. Monte Carlo minimizes the mean squared error with respect to observed returns, while TD learning converges to the dynamic programming solution.

It is highlighted that while both methods converge to the correct value asymptotically under certain assumptions, with finite data, they may converge to different values. Monte Carlo minimizes the mean squared error with respect to observed returns, while TD learning converges to the dynamic programming solution. (1h17m32s)

The discussion involves using a maximum likelihood model for policy evaluation in a Markov Decision Process (MDP), highlighting the concept of certainty equivalence. The result from a batch process using TD(0) is equivalent to computing a maximum likelihood MDP from the data and applying dynamic programming.

The discussion involves using a maximum likelihood model for policy evaluation in a Markov Decision Process (MDP), highlighting the concept of certainty equivalence. The result from a batch process using TD(0) is equivalent to computing a maximum likelihood MDP from the data and applying dynamic programming. (1h18m5s)

Temporal Difference (TD) learning explicitly uses the Markov assumption to relate the value of state A to state B, incorporating the immediate reward and future states. This contrasts with Monte Carlo methods, which aim to minimize mean squared error without explicitly using the Markov structure.

Temporal Difference (TD) learning explicitly uses the Markov assumption to relate the value of state A to state B, incorporating the immediate reward and future states. This contrasts with Monte Carlo methods, which aim to minimize mean squared error without explicitly using the Markov structure. (1h18m42s)

The choice between TD learning and Monte Carlo methods depends on whether the Markov property is satisfied, as they can yield different solutions. TD learning exploits the Markov structure, which can be beneficial for estimating earlier states.

The choice between TD learning and Monte Carlo methods depends on whether the Markov property is satisfied, as they can yield different solutions. TD learning exploits the Markov structure, which can be beneficial for estimating earlier states. (1h19m29s)

The session concludes with a summary of policy evaluation in tabular settings, and a preview of upcoming topics, including control and function approximation.

The session concludes with a summary of policy evaluation in tabular settings, and a preview of upcoming topics, including control and function approximation. (1h19m46s)

Stanford CS234 Reinforcement Learning I Policy Evaluation I 2024 I Lecture 3

A poll is conducted to refresh understanding of Markov Decision Processes (MDPs), focusing on their properties and guarantees.

Tabular MDPs are discussed, where the value of each state can be represented in a table, contrasting with neural networks that do not have one parameter per state.

The concept of (a^s) is explained as the number of possible deterministic policies, where (a) is the number of actions and (s) is the number of states.

It is confirmed that value iteration and policy iteration both asymptotically converge and compute the correct value function in tabular discrete MDPs.

A discussion is held on the convergence of value iteration, which is guaranteed if the discount factor (\gamma) is less than one, although it may require more iterations compared to policy iteration.

An example is given to illustrate the difference between policy iteration and value iteration, using a simple MDP with one state and one action, where policy iteration takes one round, but value iteration continues until the value function stabilizes.

The discussion involves using a geometric series to evaluate the value function in reinforcement learning, where the value of a state is calculated as (1/(1-\gamma)), with (\gamma) being the discount factor. This reflects the idea of staying in the same state and taking the same action repeatedly.

The focus is on model-free policy evaluation in a simple setting without function approximation, where the agent learns by trying actions in the world without a given dynamics or reward model. This is done in a tabular setting where values for each state can be written separately.

Logistics for the course include office hours listed on a Google Calendar, with specific times for project and conceptual questions. Students are encouraged to discuss projects, which can involve new ideas, applications, or replicating existing research in reinforcement learning.

The session aims to explore how to learn through direct interaction with the environment, continuing the theme of policy evaluation.

The discussion focuses on evaluating the effectiveness of decisions under a fixed policy, using data from the environment obtained by executing that policy. This approach is crucial for making decisions and improving policies in complex algorithms like deep Q-learning and policy gradient methods.

The session aims to cover topics such as multicolor policy evaluation, temporal difference learning, certainty equivalence, and batch policy evaluation. Temporal difference learning and Q-learning are highlighted, with Q-learning being the control version of temporal difference learning.

Definitions are provided for key concepts: the return (G) as the discounted sum of rewards from a state, the state value function as the average reward, and the state-action value as the expected discounted sum of rewards when starting from a state and taking a specific action.

Dynamic programming for policy evaluation is discussed, particularly when models are available, using Bellman-like backups for a specific policy. This differs from the Bellman equation as it does not involve maximizing over actions but follows the policy's specified actions.

The concept of bootstrapping is explained, where an estimate of the value function is used to calculate the expected discounted sum of future rewards. This involves using an immediate reward plus the discounted sum of future rewards as an estimate for further calculations.

Monte Carlo policy evaluation is a straightforward and widely used method in reinforcement learning, where the idea is to simulate or act in the real world using a given policy to determine the value function by averaging the returns from multiple episodes.

The focus is primarily on deterministic policies, which simplifies the process by avoiding the need to calculate expectations over actions. The value function is essentially the mean of the returns obtained by following the policy.

An example of Monte Carlo policy evaluation is assessing the average effectiveness of a medical treatment by applying it to multiple patients and averaging their outcomes. This method involves executing the policy across various episodes and averaging the results.

The method can handle cases where trajectories have different lengths, such as when patients drop out of a trial or complete their treatment at different times. The value function is derived by averaging over all trajectories, regardless of their length.

Monte Carlo policy evaluation does not assume that the system is a Markov decision process (MDP), meaning it can be applied even if the system's state is not fully described by the available features. It simply involves averaging the outcomes of multiple policy rollouts.

This approach is applicable only to episodic MDPs, where episodes have a defined end, allowing for the calculation of total returns. It is not suitable for continuous, never-ending processes without additional approximations.

The discussion addresses whether a treatment can be considered episodic if the episodes have different lengths, concluding that it can be episodic as long as the episode is guaranteed to end, allowing for averaging over outcomes.

To evaluate the value function for all states, it is necessary to observe all states within the trajectories, and methods for estimating these values are discussed.

The process involves initializing variables to count the number of times a state has been updated and to track the returns from each state, using the every-visit Monte Carlo method to sample episodes and compute the discounted sum of rewards.

The first-visit Monte Carlo evaluation method updates a state at most once per episode, even if the state is visited multiple times, which can be problematic if a state is rarely visited, leading to undefined values for such states.

Estimating rare side effects of treatment plans, such as those from the COVID vaccine, requires a large number of trajectories because some side effects only appear after many observations. This highlights the challenge of needing numerous episodes to observe infrequent states.

In reinforcement learning, the "every visit" method updates the value estimate every time a state is encountered in a trajectory, which can be beneficial for long trajectories where a state is visited multiple times.

The "incremental Monte Carlo" method maintains a running estimate of the value under a policy for a particular state, updating it smoothly as more data is collected. This involves weighing the old estimate by the number of times the state has been visited and incorporating the new return observed.

The incremental update process is similar to a learning rate in machine learning, where the updated value is a combination of the old value and the new data. This method allows for a flexible learning rate, denoted as Alpha, to update the value function for a state.

The estimates used in these methods, often referred to as targets, involve calculating the return starting from a specific state until the end of the episode. This concept is visually similar to structures like Mini Max trees or Expecting Max trees, which are used in algorithms such as AlphaGo.

Policy evaluation is conceptualized as a tree structure, where each state leads to a fixed action, and the branching factor represents all possible next states. The process involves calculating the expected value over all potential future states and propagating these values back to the root.

The aim is to compute the value function in a computationally and sample-efficient manner. Monte Carlo policy evaluation is introduced as a method to approximate full expectations by using samples of returns, which are averaged over many iterations to estimate the value function.

Computational efficiency refers to the cost of computation and memory usage, while statistical efficiency relates to how the accuracy of the estimate changes with the amount of data. Empirical accuracy, such as mean squared error, is also important in evaluating the quality of an estimator.

First-visit Monte Carlo is unbiased and consistent, as it treats all data as independent and identically distributed (IID). In contrast, every-visit Monte Carlo is biased, as it does not treat data as IID.

Temporal difference learning is introduced as a key concept in reinforcement learning, related to Q-learning, which will be covered in a subsequent lecture. It is highlighted as central and novel to reinforcement learning, according to the textbook by Sutton and Barto.

Temporal difference learning is model-free, meaning it does not require a parametric representation of the reward function or dynamics model. It can be used in both episodic settings and infinite discounted horizon settings.

The method involves updating estimates of the value of a state immediately after each state-action-reward-next state tuple, aiming to compute the expected discounted sum of rewards for a particular policy.

The approach combines sampling to approximate expectations and bootstrapping to approximate future returns. Instead of using the full return, it uses the immediate reward plus the discounted sum of future rewards, leveraging the current estimate of the value function for the next state.

In reinforcement learning, the value of the current state can be updated immediately upon reaching the next state, S Prime, without waiting until the end of the episode. This approach is applicable to infinite horizon problems.

The TD0 learning algorithm involves sampling a tuple of state, action, reward, and next state, updating the value for the starting state, and repeating this process iteratively. This is similar to Q-learning but without the use of a maximum function.

An example is provided where a policy always takes action A1, with a discount factor of one. In this scenario, any action from state S1 or S7 terminates the episode. The trajectory starts in state S3, taking action A1, receiving a reward of zero, and transitioning to state S2.

In the given example, the only state that would be updated to a non-zero value is state S1, as it provides a reward of one. This update occurs when transitioning to state S1, highlighting the importance of non-zero rewards or transitions to states with non-zero values for updates.

The value function is updated using Temporal Difference (TD) learning, which updates after every state-action-reward-next state tuple, resulting in a different value function estimate compared to Monte Carlo methods.

In TD learning, the value of a terminal state is always zero, and the update involves a combination of the current estimate and the observed reward, weighted by a factor alpha.

Monte Carlo methods wait until the end of an episode to update the value function using the returns from all visited states, which can lead to different estimates compared to TD learning.

The choice between TD and Monte Carlo methods can affect data efficiency, especially when data is limited at the beginning of learning. Sample efficiency is an important consideration in reinforcement learning.

TD learning approximates the expected value by sampling the next state and bootstraps using the current value estimate, unlike Monte Carlo methods that sample the entire value function.

The learning rate in TD learning affects how much the TD target is weighted compared to the past value estimate, and this has implications for convergence, especially in stochastic state spaces.

A deterministic Markov Decision Process (MDP) is described, where the probability of transitioning to a specific next state given a state and action is 1, meaning there is no stochasticity in the transitions.

Participants are encouraged to think about which statements are true regarding the deterministic MDP and to discuss their thoughts with others.

A discussion takes place about the implications of different values of Alpha in the context of Temporal Difference (TD) learning. If Alpha equals zero, the TD target is ignored, and no updates occur. If Alpha equals one, the estimate is completely updated with each observation.

An example is given to illustrate stochastic systems, where two states may transition to each other with certain probabilities, such as a 50% chance of receiving a reward of +1 or -1, highlighting the challenges of using a single sample to approximate expectations in stochastic environments.

It is noted that deterministic systems can still converge even if Alpha equals one, especially in cases with terminal states where no further transitions occur, such as reaching a terminal state with a fixed reward.

In cases where there is no stochasticity towards the end of an episode, convergence can occur.

Temporal Difference (TD) learning involves bootstrapping and sampling to approximate expectations, which can be used in both episodic and infinite horizon settings. It generally has lower variance and is a consistent estimator if the learning rate conditions are met.

TD(0) is introduced, which involves taking the immediate reward and bootstrapping by plugging in the value of the next state, as opposed to summing all discounted rewards for the entire episode.

There are various TD methods that interpolate between taking one step and plugging in the value versus not using any value function approximation.

The choice between computational complexity and performance often depends on the domain, and there is flexibility in using methods that balance these factors.

Some methods do not require the Markov assumption, allowing for flexibility in systems where this assumption may not hold, often resulting in lower bias.

The discussion also relates these ideas to dynamic programming, which can be used for policy evaluation when models are given.

Certainty equivalence approaches involve using data obtained from executing a policy to estimate a reward or dynamics model, such as a maximum likelihood Markov Decision Process (MDP) model, especially in a tabular setting.

The method is consistent and will converge to the correct solution for Markov models. It can be used for all policy evaluations, although it is computationally intensive. The term "NSA" refers to the number of times a state-action pair has been encountered.

The approach is different from Monte Carlo methods, as it involves generating models and propagating information rather than directly calculating returns. This distinction will lead to different decision-making processes.

Batch policy evaluation involves using a set of episodes to extract as much information as possible from the available data. This is particularly relevant in fields where data collection is expensive or potentially harmful, such as patient, student, or legal data.

The process of policy evaluation using a finite amount of data involves repeatedly sampling episodes and applying Monte Carlo or TD(0) methods to evaluate the policy. This process is iterated multiple times to understand the convergence of these methods.

An example is provided with a small domain consisting of two states, where the discount factor gamma is set to one, indicating no discounting. There are eight episodes of experience, with different trajectories and rewards observed.

In one scenario, starting in state A leads to a reward of zero, transitioning to state B with another reward of zero. In another scenario, starting in state B results in an immediate reward of one, observed six times, and once with a reward of zero.

When applying TD learning, the updates involve a combination of the old estimate and the immediate reward, with no future discounted rewards since the process terminates after state B. Iterating over this data leads to an estimated value of 0.75 for state B.

For Monte Carlo evaluation, the process involves averaging all returns starting from state B. The question is whether the Monte Carlo method will converge to the same value as TD learning. The Monte Carlo estimate also results in a value of 0.75 for state B, as it averages the observed returns.

The discussion involves evaluating the convergence of value functions ( V(a) ) using two different methods: Monte Carlo updates and Temporal Difference (TD) learning, in scenarios with finite data.

The question posed is whether these methods compute the same value when data is limited, as opposed to having infinite data where both methods can be consistent.

In the example discussed, TD learning converges to a value of 75 for ( V(a) ), based on the immediate reward being zero and the discounted reward from the next state, which was previously calculated as 75.

Monte Carlo, on the other hand, does not converge to 75; instead, it converges to zero because it averages over all observed returns, and in this case, only one trajectory was observed where ( a ) appeared, resulting in a return of zero.

Temporal Difference (TD) learning explicitly uses the Markov assumption to relate the value of state A to state B, incorporating the immediate reward and future states. This contrasts with Monte Carlo methods, which aim to minimize mean squared error without explicitly using the Markov structure.

The choice between TD learning and Monte Carlo methods depends on whether the Markov property is satisfied, as they can yield different solutions. TD learning exploits the Markov structure, which can be beneficial for estimating earlier states.

The session concludes with a summary of policy evaluation in tabular settings, and a preview of upcoming topics, including control and function approximation.

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